Tensor products of principal series for the De Sitter group

Abstract

The decomposition of the tensor product of two principal series representations is determined for the simply connected double covering, G = Spin(4, 1), of the DeSitter group. The main result is that this decomposition consists of two pieces, T, and Td, where T, is a continuous direct sum with respect to Plancherel measure on G of representations from the principal series only and Td is a discrete sum of representations from the discrete series of G. The multiplicities of representations occurring in T, and Td are all finite. Introduction. Let G = Spin(4, 1) be the simply connected double covering of the DeSitter group, G = KAN an Iwasawa decomposition of G, M the centralizer of A in K, and P = MAN the associated minimal parabolic subgroup of G. For a E M and T E A, a x T is a representation of P via a X T(man) = a(m)T(a) and a representation of the form 7(a, T) = Indp a x T is called a principal series representation of G. The main goal of this paper is to determine the decomposition of the tensor product of two principal series representations of G into irreducibles. It was shown in [7], by using Mackey's tensor product theorem and the Mackey-Anh reciprocity theorem, that this problem reduces to knowing how to decompose the restriction to MA of almost every principal series representation of G and each discrete series representation of G. For a representation v7 belonging to the principal series of G, the restriction of v7 to MA, (7)MAA was determined by using Mackey's subgroup theorem. However, in that paper, we were not able to determine explicitly (7)MA for a representation 'r belonging to the discrete series of G. This we do in ?3 of this paper by using Lie algebraic methods and the realizations of these representations given by Dixmier in [2]. This paper is organized as follows. In ?? 1 and 2 we summarize the main results concerning the structure and representation theory of G that we shall use. In ?3 we determine (7)MA when v7 is a discrete series representation of G. We also include the results of [7] concerning the decomposition of (@)MA when v7 is a principal series representation of G. In ?4 we show how to decompose the tensor product of two principal series representations of G. The main results are contained in Theorem 4. The basic methodology used in this paper to decompose principal series tensor products originates in the works of G. Mackey [6], N. Anh [1], and F. Williams [11]. Received by the editors April 15, 1980. 1980 Mathematics Subject Classification Primary 22E43, 81C40. 'This research was partially supported by a grant from the National Science Foundation. ? 1981 American Mathematical Society 0002-9947/81 /0000-0207/$04.75 121 This content downloaded from 157.55.39.118 on Fri, 22 Apr 2016 04:32:41 UTC All use subject to http://about.jstor.org/terms